Facebook Page
Twitter
RSS
+ Reply to Thread
Results 1 to 4 of 4

Thread: Four circles

  1. MHB Apprentice

    Status
    Offline
    Join Date
    Jan 2013
    Location
    Moldova
    Posts
    35
    Thanks
    13 times
    Thanked
    11 times
    #1

    The centers of three circles are situated on a line. The center of the fourth circle is situated at given distance d from that line. What is the radius of the fourth circle if we know that each circle is tangent to other three. Please give me a hint, if you can. Answer: $ \displaystyle d/2.$

  2. زيد اليافعي
    MHB Site Helper
    MHB Math Helper
    ZaidAlyafey's Avatar
    Status
    Offline
    Join Date
    Jan 2013
    Location
    KSA
    Posts
    1,666
    Thanks
    3,671 times
    Thanked
    3,865 times
    Thank/Post
    2.320
    Awards
    MHB Math Notes Award (2014)  

MHB Best Ideas (2014)  

MHB Best Ideas (Jul-Dec 2013)  

MHB Analysis Award (Jul-Dec 2013)  

MHB Calculus Award (Jul-Dec 2013)
    #2
    It is not clear what is d from the picture .

  3. MHB Craftsman
    mathmaniac's Avatar
    Status
    Offline
    Join Date
    Mar 2013
    Posts
    188
    Thanks
    167 times
    Thanked
    137 times
    #3
    Quote Originally Posted by ZaidAlyafey View Post
    It is not clear what is d from the picture .
    Quote Originally Posted by Andrei View Post
    The center of the fourth circle is situated at given distance d from that line.
    I am also interested in the solution.

    Quote Originally Posted by Andrei View Post
    The center of the fourth circle is situated at given distance d from that line.
    I am also interested in the solution.

    Edit:

    I also don't know where to start with this one.

    But I really wonder why this works for all proportions in which the centre of the biggest circle is divided into.

    It looks like d/2 is the radius of the 4th circle for all proportions (by my visualisation).

    But to start working on it,I think I have to get to paper and pencil.

    At first I would be looking for the case in which the centre of the big circle is divided into two equal parts as I assume it is easier to understand.
    Last edited by mathmaniac; April 10th, 2013 at 23:22. Reason: I was asked to provide more information.

  4. MHB Oldtimer
    MHB Site Helper
    MHB Math Scholar
    Opalg's Avatar
    Status
    Offline
    Join Date
    Feb 2012
    Location
    Leeds, UK
    Posts
    2,093
    Thanks
    704 times
    Thanked
    5,901 times
    Thank/Post
    2.819
    Awards
    Graduate POTW Award (2016)  

MHB Analysis Award (2016)  

Graduate POTW Award (2015)  

Graduate POTW Award (Jul-Dec 2013)  

MHB Pre-University Math Award (Jul-Dec 2013)
    #4
    Quote Originally Posted by Andrei View Post
    The centers of three circles are situated on a line. The center of the fourth circle is situated at given distance d from that line. What is the radius of the fourth circle if we know that each circle is tangent to other three. Please give me a hint, if you can. Answer: $ \displaystyle d/2.$
    Write $r_1,\ r_2,\ r_3,\ r_4$ for the radii of the circles centred at $O_1,\ O_2,\ O_3,\ O_4$ respectively. Notice that $r_1 = r_2+r_3$. To calculate $r_4$ in terms of $r_2$ and $r_3$, use . For that, you need to use the curvatures $k_i = \pm1/r_i$ $(i=1,2,3,4)$. The first of these, $k_1$, will be negative (because the large circle touches the other three internally). Descartes' theorem says that $k_4 = k_1+k_2+k_3 \pm\sqrt{k_1k_2 + k_2k_3 + k_3k_1}$. When you substitute $k_1 = -1/(r_2+r_3)$, $k_i = 1/r_i$ for $i=2,3,4$, you find that the part inside the square root sign is zero (this corresponds to the fact that the centres $O_1,\ O_2,\ O_3$ are collinear). That gives you a formula for $r_4$ in terms of $r_2$ and $r_3$.

    Now you have to bring in the distance $d$. I think the neatest way to do that is to use the fact that the area of triangle $O_2O_3O_4$ is $\frac12d(r_2+r_3)$. It is also given by Heron's formula in terms of $r_2,\ r_3,\ r_4$. Compare the two results and you will find that $r_4 = \frac12d$.
    Last edited by Opalg; April 11th, 2013 at 13:54.

Similar Threads

  1. Two different circles in the plane with nonempty intersection
    By Arnold in forum Discrete Mathematics, Set Theory, and Logic
    Replies: 3
    Last Post: January 11th, 2013, 14:27
  2. Two circles intersecting, a lot of lines.
    By caffeinemachine in forum Geometry
    Replies: 8
    Last Post: December 13th, 2012, 17:09
  3. Another problem on circles
    By DrunkenOldFool in forum Geometry
    Replies: 2
    Last Post: November 8th, 2012, 09:21
  4. Circles
    By DrunkenOldFool in forum Geometry
    Replies: 6
    Last Post: November 6th, 2012, 09:54
  5. Circles
    By sat in forum Geometry
    Replies: 7
    Last Post: February 6th, 2012, 22:12

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •  
Math Help Boards